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A certain loan program offers an interest rate of 5% per year, compounded continuously. Assuming no payments are made, how much would be owed after four years on a loan of $3900?

Do not round any intermediate computations, and round your answer to the nearest cent.


Sagot :

To determine the amount owed after four years on a loan of [tex]$3900 at an interest rate of 5% per year, compounded continuously, we will use the formula for continuous compounding of interest: \[ A = P \cdot e^{rt} \] where: - \( A \) is the amount owed after time \( t \), - \( P \) is the principal, or initial amount of the loan, - \( e \) is the base of the natural logarithm, approximately equal to 2.71828, - \( r \) is the annual interest rate (expressed as a decimal), - \( t \) is the time the money is invested or borrowed for, in years. Given: - The principal \( P \) is $[/tex]3900,
- The annual interest rate [tex]\( r \)[/tex] is 5%, or 0.05 as a decimal,
- The time [tex]\( t \)[/tex] is 4 years.

Let's plug these values into the formula:

[tex]\[ A = 3900 \cdot e^{0.05 \cdot 4} \][/tex]

First, calculate the exponent:

[tex]\[ 0.05 \cdot 4 = 0.20 \][/tex]

Now, we need to calculate [tex]\( e^{0.20} \)[/tex]. Using the value:

[tex]\[ e^{0.20} \approx 1.22140275816 \][/tex]

Then, multiply this value by the principal [tex]\( 3900 \)[/tex]:

[tex]\[ A = 3900 \cdot 1.22140275816 \approx 4763.470756824662 \][/tex]

Next, we round the amount to the nearest cent. The result is:

[tex]\[ A \approx 4763.47 \][/tex]

Therefore, after four years, the amount owed on a loan of [tex]$3900 with an annual interest rate of 5%, compounded continuously, would be approximately $[/tex]4763.47.
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